Pre-Algebra Examples

$\frac{8}{5 r - 1} - 8 \div \frac{8}{5 r - 1} + 8$

Step 1

Simplify each term.

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Step 1.1

To divide by a fraction, multiply by its reciprocal.

$\frac{8}{5 r - 1} - (8 \frac{5 r - 1}{8}) + 8$

Step 1.2

Cancel the common factor of $8$ .

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Step 1.2.1

Cancel the common factor.

$\frac{8}{5 r - 1} - (8 \frac{5 r - 1}{8}) + 8$

Step 1.2.2

Rewrite the expression.

$\frac{8}{5 r - 1} - (5 r - 1) + 8$

Step 1.3

Apply the distributive property.

$\frac{8}{5 r - 1} - (5 r) - - 1 + 8$

Step 1.4

Multiply $5$ by $- 1$ .

$\frac{8}{5 r - 1} - 5 r - - 1 + 8$

Step 1.5

Multiply $- 1$ by $- 1$ .

$\frac{8}{5 r - 1} - 5 r + 1 + 8$

Step 2

Find the common denominator.

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Step 2.1

Write $- 5 r$ as a fraction with denominator $1$ .

$\frac{8}{5 r - 1} + \frac{- 5 r}{1} + 1 + 8$

Step 2.2

Multiply $\frac{- 5 r}{1}$ by $\frac{5 r - 1}{5 r - 1}$ .

$\frac{8}{5 r - 1} + \frac{- 5 r}{1} \cdot \frac{5 r - 1}{5 r - 1} + 1 + 8$

Step 2.3

Multiply $\frac{- 5 r}{1}$ by $\frac{5 r - 1}{5 r - 1}$ .

$\frac{8}{5 r - 1} + \frac{- 5 r (5 r - 1)}{5 r - 1} + 1 + 8$

Step 2.4

Write $1$ as a fraction with denominator $1$ .

$\frac{8}{5 r - 1} + \frac{- 5 r (5 r - 1)}{5 r - 1} + \frac{1}{1} + 8$

Step 2.5

Multiply $\frac{1}{1}$ by $\frac{5 r - 1}{5 r - 1}$ .

$\frac{8}{5 r - 1} + \frac{- 5 r (5 r - 1)}{5 r - 1} + \frac{1}{1} \cdot \frac{5 r - 1}{5 r - 1} + 8$

Step 2.6

Multiply $\frac{1}{1}$ by $\frac{5 r - 1}{5 r - 1}$ .

$\frac{8}{5 r - 1} + \frac{- 5 r (5 r - 1)}{5 r - 1} + \frac{5 r - 1}{5 r - 1} + 8$

Step 2.7

Write $8$ as a fraction with denominator $1$ .

$\frac{8}{5 r - 1} + \frac{- 5 r (5 r - 1)}{5 r - 1} + \frac{5 r - 1}{5 r - 1} + \frac{8}{1}$

Step 2.8

Multiply $\frac{8}{1}$ by $\frac{5 r - 1}{5 r - 1}$ .

$\frac{8}{5 r - 1} + \frac{- 5 r (5 r - 1)}{5 r - 1} + \frac{5 r - 1}{5 r - 1} + \frac{8}{1} \cdot \frac{5 r - 1}{5 r - 1}$

Step 2.9

Multiply $\frac{8}{1}$ by $\frac{5 r - 1}{5 r - 1}$ .

$\frac{8}{5 r - 1} + \frac{- 5 r (5 r - 1)}{5 r - 1} + \frac{5 r - 1}{5 r - 1} + \frac{8 (5 r - 1)}{5 r - 1}$

Step 3

Combine the numerators over the common denominator.

$\frac{8 - 5 r (5 r - 1) + 5 r - 1 + 8 (5 r - 1)}{5 r - 1}$

Step 4

Simplify each term.

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Step 4.1

Apply the distributive property.

$\frac{8 - 5 r (5 r) - 5 r \cdot - 1 + 5 r - 1 + 8 (5 r - 1)}{5 r - 1}$

Step 4.2

Rewrite using the commutative property of multiplication.

$\frac{8 - 5 \cdot 5 r \cdot r - 5 r \cdot - 1 + 5 r - 1 + 8 (5 r - 1)}{5 r - 1}$

Step 4.3

Multiply $- 1$ by $- 5$ .

$\frac{8 - 5 \cdot 5 r \cdot r + 5 r + 5 r - 1 + 8 (5 r - 1)}{5 r - 1}$

Step 4.4

Simplify each term.

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Step 4.4.1

Multiply $r$ by $r$ by adding the exponents.

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Step 4.4.1.1

Move $r$ .

$\frac{8 - 5 \cdot 5 (r \cdot r) + 5 r + 5 r - 1 + 8 (5 r - 1)}{5 r - 1}$

Step 4.4.1.2

Multiply $r$ by $r$ .

$\frac{8 - 5 \cdot 5 r^{2} + 5 r + 5 r - 1 + 8 (5 r - 1)}{5 r - 1}$

Step 4.4.2

Multiply $- 5$ by $5$ .

$\frac{8 - 25 r^{2} + 5 r + 5 r - 1 + 8 (5 r - 1)}{5 r - 1}$

Step 4.5

Apply the distributive property.

$\frac{8 - 25 r^{2} + 5 r + 5 r - 1 + 8 (5 r) + 8 \cdot - 1}{5 r - 1}$

Step 4.6

Multiply $5$ by $8$ .

$\frac{8 - 25 r^{2} + 5 r + 5 r - 1 + 40 r + 8 \cdot - 1}{5 r - 1}$

Step 4.7

Multiply $8$ by $- 1$ .

$\frac{8 - 25 r^{2} + 5 r + 5 r - 1 + 40 r - 8}{5 r - 1}$

Step 5

Simplify terms.

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Step 5.1

Combine the opposite terms in $8 - 25 r^{2} + 5 r + 5 r - 1 + 40 r - 8$ .

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Step 5.1.1

Subtract $8$ from $8$ .

$\frac{- 25 r^{2} + 5 r + 5 r - 1 + 40 r + 0}{5 r - 1}$

Step 5.1.2

Add $- 25 r^{2} + 5 r + 5 r - 1 + 40 r$ and $0$ .

$\frac{- 25 r^{2} + 5 r + 5 r - 1 + 40 r}{5 r - 1}$

Step 5.2

Add $5 r$ and $5 r$ .

$\frac{- 25 r^{2} + 10 r - 1 + 40 r}{5 r - 1}$

Step 5.3

Add $10 r$ and $40 r$ .

$\frac{- 25 r^{2} + 50 r - 1}{5 r - 1}$

Step 5.4

Factor $- 1$ out of $- 25 r^{2}$ .

$\frac{- (25 r^{2}) + 50 r - 1}{5 r - 1}$

Step 5.5

Factor $- 1$ out of $50 r$ .

$\frac{- (25 r^{2}) - (- 50 r) - 1}{5 r - 1}$

Step 5.6

Factor $- 1$ out of $- (25 r^{2}) - (- 50 r)$ .

$\frac{- (25 r^{2} - 50 r) - 1}{5 r - 1}$

Step 5.7

Rewrite $- 1$ as $- 1 (1)$ .

$\frac{- (25 r^{2} - 50 r) - 1 (1)}{5 r - 1}$

Step 5.8

Factor $- 1$ out of $- (25 r^{2} - 50 r) - 1 (1)$ .

$\frac{- (25 r^{2} - 50 r + 1)}{5 r - 1}$

Step 5.9

Simplify the expression.

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Step 5.9.1

Rewrite $- (25 r^{2} - 50 r + 1)$ as $- 1 (25 r^{2} - 50 r + 1)$ .

$\frac{- 1 (25 r^{2} - 50 r + 1)}{5 r - 1}$

Step 5.9.2

Move the negative in front of the fraction.

$- \frac{25 r^{2} - 50 r + 1}{5 r - 1}$